Богомолов Федор Алексеевич

Unirationality and existence of infinitely transitive models

2013 · CHAPTER · en

We study unirational algebraic varieties and the fields of rational functions on them. We show that after adding a finite number of variables some of these fields admit an infinitely transitive model. The latter is an algebraic variety with the given field of rational functions and an infinitely transitive regular action of a group of algebraic automorphisms generated by unipotent algebraic subgroups. We expect that this property holds for all unirational varieties and in fact is a peculiar one for this class of algebraic varieties among those varieties which are rationally connected.

Isoclinism and Stable Cohomology of Wreath Products

2013 · CHAPTER · en

Using the notion of isoclinism introduced by P. Hall for finite p-groups, we show that many important classes of finite p-groups have stable cohomology detected by abelian subgroups (see Theorem 11). Moreover, we show that the stable cohomology of the n-fold wreath product Gn=Z/p≀…≀Z/p of cyclic groups Z/p is detected by elementary abelian p-subgroups and we describe the resulting cohomology algebra explicitly. Some applications to the computation of unramified and stable cohomology of finite groups of Lie type are given.

On stable conjugacy of finite subgroups of the plane Cremona group, I

2013 · ARTICLE · en

We discuss the problem of stable conjugacy of finite subgroups of Cremona groups. We show that the group $H^1(G,Pic(X))$ is a stable birational invariant and compute this group in some cases.

Closed symmetric 2-differentials of the 1st kind

2013 · PREPRINT · en

A closed symmetric differential of the 1st kind is a differential that locally is the product of closed holomorphic 1-forms. We show that closed symmetric 2-differentials of the 1st kind on a projective manifold $X$ come from maps of $X$ to cyclic or dihedral quotients of Abelian varieties and that their presence implies that the fundamental group of $X$ (case of rank 2) or of the complement $X\setminus E$ of a divisor $E$ with negative properties (case of rank 1) contains subgroup of finite index with infinite abelianization. Other results include: i) the identification of the differential operator characterizing closed symmetric 2-differentials on surfaces (which provides in this case a connection to flat Riemannian metrics) and ii) projective manifolds $X$ having symmetric 2-differentials $w$ that are the product of two closed meromorphic 1-forms are irregular, in fact if $w$ is not of the 1st kind (which can happen), then $X$ has a fibration $f:X \to C$ over a curve of genus $\ge 1$.

On uniformly rational varieties

2013 · PREPRINT · en

We investigate basic properties of uniformly rational varieties, i.e. those smooth varieties for which every point has a Zariski open neighborhood isomorphic to an open subset of A^n. It is an open question of Gromov whether all smooth rational varieties are uniformly rational. We discuss some potential criteria that might allow one to show that they form a proper subclass in the class of all smooth rational varieties. Finally we prove that small algebraic resolutions and big resolutions of nodal cubic threefolds are uniformly rational.

Stable cohomology of alternating groups

2012 · PREPRINT · en

In this article we determine the stable cohomology groups H^i_s (A_n, Z/p) of the alternating groups A_n for all integers n and i, and all primes p.

Collineation group as a subgroup of the symmetric group

2012 · PREPRINT · en

Let $\Psi$ be the projectivization (i.e., the set of one-dimensional vector subspaces) of a vector space of dimension $\ge 3$ over a field. Let $H$ be a closed (in the pointwise convergence topology) subgroup of the permutation group $\mathfrak{S}_{\Psi}$ of the set $\Psi$. Suppose that $H$ contains the projective group and an arbitrary self-bijection of $\Psi$ transforming a triple of collinear points to a non-collinear triple. It is well-known from \cite{KantorMcDonough} that if $\Psi$ is finite then $H$ contains the alternating subgroup $\mathfrak{A}_{\Psi}$ of $\mathfrak{S}_{\Psi}$. We show in Theorem \ref{density} below that $H=\mathfrak{S}_{\Psi}$, if $\Psi$ is infinite.

Unirationality and existence of infinitely transitive models

2012 · PREPRINT · en

We study unirational algebraic varieties and the fields of rational functions on them. We show that after adding a finite number of variables some of these fields admit an \emph{infinitely transitive model}. The latter is an algebraic variety with the given field of rational functions and an infinitely transitive regular action of an algebraic group generated by unipotent algebraic subgroups. We expect this property holds for all unirational varieties and in fact is a peculiar one for this class of algebraic varieties among those varieties which are rationally connected.

Linear bounds for levels of stable rationality

2012 · ARTICLE · en

Let G be one of the groups SL n(ℂ), Sp 2n(ℂ), SO m(ℂ), O m(ℂ), or G 2. For a generically free G-representation V, we say that N is a level of stable rationality for V/G if V/G × ℙ N is rational. In this paper we improve known bounds for the levels of stable rationality for the quotients V/G. In particular, their growth as functions of the rank of the group is linear for G being one of the classical groups.

Isoclinism and stable cohomology of wreath products

2012 · PREPRINT · en